| L(s) = 1 | + 2.84e3·2-s + 5.99e6·4-s − 3.10e6·5-s + 3.63e8·7-s + 1.10e10·8-s − 8.84e9·10-s − 1.45e10·11-s + 1.13e11·13-s + 1.03e12·14-s + 1.89e13·16-s + 8.58e12·17-s − 2.92e13·19-s − 1.86e13·20-s − 4.14e13·22-s + 1.55e14·23-s − 4.67e14·25-s + 3.22e14·26-s + 2.17e15·28-s − 2.40e15·29-s + 2.23e15·31-s + 3.06e16·32-s + 2.44e16·34-s − 1.12e15·35-s − 3.07e16·37-s − 8.30e16·38-s − 3.44e16·40-s + 1.03e17·41-s + ⋯ |
| L(s) = 1 | + 1.96·2-s + 2.85·4-s − 0.142·5-s + 0.486·7-s + 3.64·8-s − 0.279·10-s − 0.169·11-s + 0.228·13-s + 0.954·14-s + 4.30·16-s + 1.03·17-s − 1.09·19-s − 0.406·20-s − 0.332·22-s + 0.784·23-s − 0.979·25-s + 0.447·26-s + 1.38·28-s − 1.05·29-s + 0.490·31-s + 4.80·32-s + 2.02·34-s − 0.0692·35-s − 1.05·37-s − 2.14·38-s − 0.519·40-s + 1.20·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(7.109010408\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.109010408\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 711 p^{2} T + p^{21} T^{2} \) |
| 5 | \( 1 + 124398 p^{2} T + p^{21} T^{2} \) |
| 7 | \( 1 - 51900560 p T + p^{21} T^{2} \) |
| 11 | \( 1 + 1325621196 p T + p^{21} T^{2} \) |
| 13 | \( 1 - 113350790702 T + p^{21} T^{2} \) |
| 17 | \( 1 - 505258211646 p T + p^{21} T^{2} \) |
| 19 | \( 1 + 1536996803884 p T + p^{21} T^{2} \) |
| 23 | \( 1 - 155899214954280 T + p^{21} T^{2} \) |
| 29 | \( 1 + 2400788707090758 T + p^{21} T^{2} \) |
| 31 | \( 1 - 2239820676947000 T + p^{21} T^{2} \) |
| 37 | \( 1 + 30785069383298890 T + p^{21} T^{2} \) |
| 41 | \( 1 - 103207571041281030 T + p^{21} T^{2} \) |
| 43 | \( 1 + 165557270617488124 T + p^{21} T^{2} \) |
| 47 | \( 1 - 66587216226477408 T + p^{21} T^{2} \) |
| 53 | \( 1 + 435422766592881630 T + p^{21} T^{2} \) |
| 59 | \( 1 + 5534365798259081316 T + p^{21} T^{2} \) |
| 61 | \( 1 + 7176205164722961202 T + p^{21} T^{2} \) |
| 67 | \( 1 + 15755449453068299812 T + p^{21} T^{2} \) |
| 71 | \( 1 + 26457854874259376232 T + p^{21} T^{2} \) |
| 73 | \( 1 - 13471249335464801450 T + p^{21} T^{2} \) |
| 79 | \( 1 + 16886125085525986840 T + p^{21} T^{2} \) |
| 83 | \( 1 - \)\(17\!\cdots\!72\)\( T + p^{21} T^{2} \) |
| 89 | \( 1 - \)\(31\!\cdots\!86\)\( T + p^{21} T^{2} \) |
| 97 | \( 1 - \)\(94\!\cdots\!18\)\( T + p^{21} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.44541243290921593737038989721, −14.44920245612142903557464689293, −13.19246029078025916833384099554, −11.96565407990803606952560186919, −10.73611087579463846148801264150, −7.63113534940665496266616984154, −6.03526718926988049921880519588, −4.71313088445062717712746740429, −3.35211043474268330135376328149, −1.76707195027819258339075285549,
1.76707195027819258339075285549, 3.35211043474268330135376328149, 4.71313088445062717712746740429, 6.03526718926988049921880519588, 7.63113534940665496266616984154, 10.73611087579463846148801264150, 11.96565407990803606952560186919, 13.19246029078025916833384099554, 14.44920245612142903557464689293, 15.44541243290921593737038989721
